We consider the Cauchy problem for the velocity-discrete Boltzmann equations in any dimension. The velocity-discrete Boltzmann equations can be considered as approximations of the original velocitycontinuous Boltzmann equation. The particles move with velocities in a given finite set. We prove global in time existence of mild solutions to the Cauchy problem for initial data with finite mass, entropy and entropy production. The proof is based on the introduction of sets of particular characteristics and a new strong compactness property of the integrated collision frequency. The sets of particular characteristics are constructed in order that the density and its associated collision frequency are bounded from above. Their complements are of arbitrarily small measure. The strong compactness property of the integrated collision frequency is based on the Kolmogorov-Riesz theorem. This replaces the compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete.